2002. The consequences of spatial structure for the design and analysis of ecological field surveys. -Ecography 25: 601-615.In ecological field surveys, observations are gathered at different spatial locations. The purpose may be to relate biological response variables (e.g., species abundances) to explanatory environmental variables (e.g., soil characteristics). In the absence of prior knowledge, ecologists have been taught to rely on systematic or random sampling designs. If there is prior knowledge about the spatial patterning of the explanatory variables, obtained from either previous surveys or a pilot study, can we use this information to optimize the sampling design in order to maximize our ability to detect the relationships between the response and explanatory variables? The specific questions addressed in this paper are: a) What is the effect (type I error) of spatial autocorrelation on the statistical tests commonly used by ecologists to analyse field survey data? b) Can we eliminate, or at least minimize, the effect of spatial autocorrelation by the design of the survey? Are there designs that provide greater power for surveys, at least under certain circumstances? c) Can we eliminate or control for the effect of spatial autocorrelation during the analysis? To answer the last question, we compared regular regression analysis to a modified t-test developed by Dutilleul for correlation coefficients in the presence of spatial autocorrelation. Replicated surfaces (typically, 1000 of them) were simulated using different spatial parameters, and these surfaces were subjected to different sampling designs and methods of statistical analysis. The simulated surfaces may represent, for example, vegetation response to underlying environmental variation. This allowed us 1) to measure the frequency of type I error (the failure to reject the null hypothesis when in fact there is no effect of the environment on the response variable) and 2) to estimate the power of the different combinations of sampling designs and methods of statistical analysis (power is measured by the rate of rejection of the null hypothesis when an effect of the environment on the response variable has been created). Our results indicate that: 1) Spatial autocorrelation in both the response and environmental variables affects the classical tests of significance of correlation or regression coefficients. Spatial autocorrelation in only one of the two variables does not affect the test of significance. 2) A broad-scale spatial structure present in data has the same effect on the tests as spatial autocorrelation. When such a structure is present in one of the variables and autocorrelation is found in the other, or in both, the tests of significance have inflated rates of type I error. 3) Dutilleul's modified t-test for the correlation coefficient, corrected for spatial autocorrelation, effectively corrects for spatial autocorrelation in the data. It also effectively corrects for the presence of deterministic structures, with or without spatial aut...

A large number of methods for the analysis of the spatial structure of natural phenomena (for example, the clumping or overdispersion of tree stems, the positions of veins of ore in a rock formation, the arrangement of habitat patches in a landscape, and so on) have been developed in a wide range of scientific fields. This paper reviews many of the methods and describes the relationships among them, both mathematically, using the cross‐product as a unifying principle, and conceptually, based on the form of a moving window or template used in calculation. The relationships among these methods suggest that while no single method can reveal all the important characteristics of spatial data, the results of different analyses are not expected to be completely independent of each other.

The main aim of this study was to assess the ability of simple geometric measures of thunderstorm rainfall in explaining the runoff response from the watershed. For calculation of storm geometric properties (e.g. areal coverage of storm, areal coverage of the high-intensity portion of the storm, position of storm centroid and the movement of storm centroid in time), spatial information of rainfall is needed. However, generally the rainfall data consists of rainfall depth values over an unevenly spaced network of raingauges. For this study, rainfall depth values were available for 91 raingauges in a watershed of about 148 km 2. There was a question about which interpolation method should be used for obtaining uniformly gridded data. Therefore, a small study was undertaken to compare cross-validation statistics and computed geometric parameters using two interpolation methods (kriging and multiquadric). These interpolation methods were used to estimate precipitation over a uniform 100 m £ 100 m grid. The cross-validation results from the two methods were generally similar and neither method consistently performed better than the other did. In view of these results we decided to use multiquadric interpolation method for the rest of the study. Several geometric measures were then computed from interpolated surfaces for about 300 storm events occurring in a 17-year period. The correlation of these computed measures with basin runoff were then observed in an attempt to assess their relative importance in basin runoff response. It was observed that the majority of the storms (observed in the study) covered the entire watershed. Therefore, it was concluded that the areal coverage of storm was not a good indicator of the amount of runoff produced. The areal coverage of the storm core (10-min intensity greater than 25 mm/h), however, was found to be a much better predictor of runoff volume and peak rate. The most important variable in runoff production was found to be the volume of the storm core. It was also observed that the position of the storm core relative to the watershed outlet becomes more important as the catchment size increases, with storms positioned in the central portion of the watershed producing more runoff than those positioned near the outlet or near the head of the watershed. This observation indicates the importance of interaction of catchment size and shape with the spatial storm structure in runoff generation. Antecedent channel wetness was found to be of some importance in explaining runoff for the largest of the three

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Geostatistics provides tools to model, estimate, map, and eventually predict spatial patterns of tree size and growth. Variogram models and kriged maps were used to study spatial dependence of stem diameter (DBH), basal area (BA), and 10-year periodic basal area increment (BAI) in an old-growth forest stand. Temporal variation of spatial patterns was evaluated by fitting spatial stochastic models at 10-year intervals, from 1920 to 1990. The study area was a naturally seeded stand of southwestern ponderosa pine (Pinusponderosa Dougl. ex Laws. var. scopulorum) where total BA and tree density have steadily increased over the last decades. Our objective was to determine if increased stand density simply reduced individual growth rates or if it also altered spatial interactions among trees. Despite increased crowding, stem size maintained the same type of spatial dependence from 1920 to 1990. An isotropic Gaussian variogram was the model of choice to represent spatial dependence at all times. Stem size was spatially autocorrelated over distances no greater than 30 m, a measure of average patch diameter in this forest ecosystem. Because patch diameter remained constant through time, tree density increased by increasing the number of pine groups, not their horizontal dimension. Spatial dependence of stem size (DBH and BA) was always much greater and decreased less through time than that of stem increment (BAI). Spatial dependence of BAI was close to zero in the most recent decade, indicating that growth rates in 1980–1990 varied regardless of mutual tree position. Increased tree crowding corresponded not only to lower average and variance of individual growth rates, but also to reduced spatial dependence of BAI. Because growth variation was less affected by intertree distance with greater local crowding, prediction of individual growth rates benefits from information on horizontal stand structure only if tree density does not exceed threshold values. Simulation models and area estimates of tree performance in old-growth forests may be improved by including geostatistical components to summarize ecological spatial dependence.

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The interpolation of spatial data has been considered in many different forms. The various forms of kriging are among the best known in the earth sciences although techniques such as inverse distance weighting were and are in use for spatially located data. In the numerical analysis literature various forms of splines and more recently radial basis functions have been developed and used. Because these techniques have been developed in very different contexts the relationship between them has not always been apparent. Various forms of kriging are considered as well as kernel estimators, splines and radial basis functions. By using the dual form of kriging and the positive definiteness property of the variogram connections are shown between splines, kriging and radial basis functions. One of the distinctions between kriging and other interpolators is the incorporation of the support of the samples and explicit estimation of linear functionals such as spatial integrals.